The GT seminar meets on Fridays
in Math. 520,
at 1:20PM.

Organizer:
Walter Neumann.

Other
area seminars. Our e-mail
list. Archive of previous semesters

### Fall 2012, Spring 2013

Date | Speaker | Title |
---|---|---|

September 7 | Organizational meeting | |

September 14/15 | No Columbia Seminar: Brock, Canary, Duchin, Futer, Kahn, Rivin, Wise, Wolpert speak at Hunter College | Hyperbolic geometry and Teichmuller theory |

September 21/22 | No Seminar: Baumslag, Bridson, Fine et al. speak at Stevens Institute and Hunter College | Group Theory on the Hudson |

September 28 | Becca Winarski | Mapping Class Groups and Covering Spaces |

October 5 Double Header |
Paul Siegel 9:40am Room 507 Aaron Smith 1:20pm |
Positive Scalar Curvature and Secondary Index Invariants A higher Riemann--Hilbert correspondence and extensions |

October 12 | Christian Zickert | Thurston's gluing equations for PGL(n,C) |

October 19 | No Seminar | |

October 26 | Rob Meyerhof | Mom 1.5 |

November 2 Postponed to Spring |
Johanna Kutluhan POSTPONED |
POSTPONED |

November 9 | Joan Birman | The curve complex has dead ends |

November 16 9:50am Room 507. NOTE TIME! (SGGT seminars are 11am and 1:20pm) | Anne Pichon | Lipschitz geometry of complex surfaces: analytic invariants and equisingularity |

November 23 | No Seminar | Thanksgiving |

November 30 | Colin Adams | Petal diagrams for knots |

December 7 | Jessica Purcell | Geodesics in compression bodies |

December 14 9:50am December 14 1:20pm |
Chris Leininger Rinat Kashaev |
Finite rigid subsets of the curve complex The quantum beta function and state integrals of Turaev-Viro type on shaped triangulations |

## Abstracts.

#### September 28.

Becca Winarski “ Mapping Class Groups and Covering Spaces ”

**Abstract**: A fundamental question of the study of mapping class groups
is finding maps between mapping class groups of different surfaces.
Given a covering space of surfaces, one may wish to relate the
mapping class groups of the two surfaces. We say a cover of S over X
has the Birman-Hilden property if there is a finite index subgroup of
the mapping class group of X that injects into the mapping class
group of S, modulo the deck transformations. In the 1970s, Birman
and Hilden proved that regular covers have the Birman-Hilden
property. We extend their work to certain irregular branched covers.
As new applications, we prove: (1) Simple covers do not have the
Birman-Hilden property and (2) All covers with at most one branch
point do have the Birman-Hilden property.

#### October 5, 9:40AM Room 507.

Paul Siegel “Positive Scalar Curvature and Secondary Index Invariants ”

**Abstract**: The Atiyah-Singer index theorem calculates algebraic invariants of elliptic operators
on manifolds in terms of topological data. One of the more mysterious applications of this
theorem is that it provides topological obstructions to the existence of Riemannian metrics on
compact spin manifolds whose scalar curvature function is everywhere positive. In this talk I
will explain how a positive scalar curvature metric on a spin manifold determines a "secondary
index invariant" in an object called the analytic structure group, and I will show how homological
calculations with the analytic structure group can be used to give nice proofs of old and new
theorems about positive scalar curvature metrics.

#### October 5, 1:20pm.

Aaron Smith “A higher Riemann--Hilbert correspondence and extensions ”

**Abstract**: I will discuss the higher Riemann--Hilbert correspondence of the
paper, arxiv 0908.2843, and then some ongoing work describing an
extension of this correspondence to a setting involving
"A_\infty-connections". The main goal of this extension is to
incorporate some natural examples into the framework of the higher
R--H correspondence, notably the free loop space with its string
product.

#### October 12.

Christian Zickert “Thurston's gluing equations for PGL(n,C) ”

**Abstract**:Thurston's gluing equations are polynomial equations invented by Thurston to explicitly compute hyperbolic structures
or, more generally, representations in PGL(2,C). This is done via so called shape coordinates. We generalize the shape
coordinates to obtain a parametrization of representations in PGL(n,C). We give applications to quantum topology, and discuss an
intriguing duality between the shape coordinates and the Ptolemy coordinates of Garoufalidis-Thurston-Zickert. The shape
coordinates and Ptolemy coordinates can be viewed as 3-dimensional analogues of the X- and A-coordinates on higher Teichmuller
spaces due to Fock and Goncharov.

#### October 26.

Rob Meyerhof “Mom 1.5 ”

**Abstract**:This talk will discuss joint work in progress with Robert Haraway and Craig Hodgson. Mom Technology has had
considerable success in proving facts about low-volume 1-cusped hyperbolic 3-manifolds. We are attempting to generalize
Mom Technology to the case of hyperbolic 3-manifolds with totally geodesic boundary. The generalization of Mom
Technology to this setting has several interesting aspects. For example, a classical Mom(2) for a 1-cusped hyperbolic
3-manifold is a certain type of handlebody structure with two 1-handles and two 2-handles. In the case of a totally
geodesic 2-holed-torus boundary, this won't work; in fact, the simplest Mom in this setting would have 1 1-handle and 2
2-handles.

#### November 9.

Joan Birman “The curve complex has dead ends”

**Abstract**: My talk will be about new joint work with Bill Menasco, on the {\it curve
complex} $\C$ of as closed surface $S$ of genus \geq 2. This simplicial
complex was introduced by Harvey in the 1970's to study mapping class groups
of surfaces, (the MCG) acts on $\C$), and developed into a major tool by
Mazur and Minsky during the past 10 years. Vertices are homotopy classes of
simple closed curves on $S$. Two vertices $X,Y$ are distance 1 apart (and
joined by an edge) when their defining curves are disjoint. It's interesting
to think about geodesics joining two vertices, when their defining curves
intersect many times. The vertex $Y$ is a {\it dead end} with respect to
$X$ if there is no extension of geodesics joining $X$ and $Y$, past $Y$,
to longer geodesics. My talk is about new pathology in $\C$: myriad
examples of dead ends (and double dead ends) in $\C$, the simplest example
being one on a surface of genus 2, with $d(X,Y)=3$. A complete picture
emerges when we find nasc for $Y$ to be a dead end with respect to $X$ and
prove that every dead end in $\C$ has depth 1.

#### November 16, 9:50 AM.

Anne Pichon “Lipschitz geometry of complex surfaces: analytic invariants and equisingularity ”

**Abstract**: The question of defining a good notion of
equisingularity of a reduced hypersurface $\mathfrak{X} \subset
\C^n$ along a non singular complex subspace $Y \subset
\mathfrak{X}$ in a neighbourhood of a point $0 \in \mathfrak{X}$
has a long history which started in 1965 with works of
Zariski. One of the central concepts introduced by Zariski is the
algebro-geometric equisingularity, called nowadays Zariski
equisingularity, which defines the equisingularity inductively on
the codimension of $Y$ in $\mathfrak{X}$ by requiring that the
reduced discriminant locus of a suitably general projection
$p\colon \mathfrak{X} \to \C^{n-1}$ be itself equisingular along
the strata $p(Y)$.

When $Y$ has codimension one in $\mathfrak{X}$, i.e., when dealing wih a family of plane curves transversal to the parameter space $Y$, it is well known that Zariski equisingularity is equivalent to the main notions of equisingularity such as Whitney conditions for the pair $(\mathfrak{X} \setminus Y,Y)$ and topological triviality. However these properties fail to be equivalent in higher codimension.

I will present a recent joint work with Walter Neumann in which we prove that in codimension $2$, for a family of hypersurfaces in $\C^3$ with isolated singularities, Zariski equisingularity is equivalent to the constancy of the family up to bilipschitz semi-algebraic homeomorphism with respect to the outer metric.

#### November 30.

Colin Adams “Petal diagrams for knots ”

**Abstract**: Knots have traditionally been investigated by
considering projections with crossings where two strands of the knot
cross one another. Here, we consider multi-crossings (or n-crossings)
where n strands of the knot cross at a single point. We show that for
each integer \ge 2, every knot has a projection made up entirely
of n-crossings, and therefore a minimal n-crossing number c_n(K). We
investigate what is known about c_n(K) and then show that for every
knot there is an n such that c_n(K) = 1. In fact, every knot has a
projection (a certain type of arc index projection) with a single
multi-crossing that looks like a daisy as in a petal
diagram. We will
consider the implications of this.

#### December 7.

Jessica Purcell “Geodesics in compression bodies ”

**Abstract**: Every knot complement admits a tunnel system:
a collection of arcs whose complement is a handlebody. If the knot
complement is hyperbolic, and the system consists of a single
unknotting tunnel, there is an open question as to whether the tunnel
will be isotopic to a geodesic. In this talk, we address the
generalization of the question to geometrically finite compression
bodies. We conjecture that when there is a single tunnel, that tunnel
will be isotopic to a geodesic. However, we show that when there are
multiple tunnels, we can select a tunnel system for which the
geodesics in their homotopy classes self-intersect. Parts of this work
are joint with Marc Lackenby, and parts with Stephan Burton.

#### December 14 9:50am

Chris Leininger “Finite rigid subsets of the curve complex ”

**Abstract**:
The mapping class group of a surface acts on the complex of
curves of that surface, which is an infinite diameter, locally infinite
simplicial complex. An important result due to Ivanov (and generalized by
Korkmaz and Luo) is that any automorphism of the curve complex of a
surface is induced by a mapping class. This has been further extended by
Irmak, Shackleton, and Behrstock-Margalit. These results say that various
qualities of simplicial map of the curve complex to itself (not assumed to
be automorphisms) are induced by mapping classes (and hence are indeed
automorphisms). In this talk, I'll describe recent work with J. Aramayona
in which we construct *finite* subcomplexes for which every simplicial
embedding into the curve complex is the restriction of an automorphism of
the entire curve complex, and hence is induced by a mapping class.

#### December 14, 1:20pm

Rinat Kashaev “The quantum beta function and state integrals of Turaev-Viro type on shaped triangulations”

**Abstract**: A shaped triangulation is a finite triangulation of an oriented
pseudo three manifold where each tetrahedron carries dihedral angles of an
ideal hyperbolic tetrahedron. I will explain a construction which associates
to each shaped triangulation a quantum partition function in the form of an
absolutely convergent state integral invariant under shaped 3-2
Pachner moves and the shape gauge transformations
generated by the total dihedral angles around internal edges through the
Neumann-Zagier Poisson bracket. Similarly to Turaev-Viro theory, the state
variables live on edges of the triangulation but take their values on the
whole real axis. The tetrahedral weight functions enjoy a manifest tetrahedral
symmetry. At least for shaped triangulations of closed 3-manifolds, up to normalization, this
partition function is conjectured to be the absolute value squared of the
partition function of the Teichmueller TQFT. This is similar to the known
relationship between the Turaev-Viro and the Witten-Reshetikhin-Turaev
invariants of three manifolds. The talk will be based on the joint work with
Feng Luo and Grigory Vartanov arXiv:1210.8393.

# Other relevant information.

## Previous semesters:

Spring 2012, Fall 2011, 2010/11, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006.## Other area seminars.

- Columbia Symplectic Geometry/Gauge Theory Seminar
- All Columbia Math Dept Seminars
- CUNY Geometry and Topology Seminar
- CUNY Complex Analysis & Dynamics Seminar
- CUNY Magnus Seminar
- Princeton Topology Seminar.

## Our e-mail list.

Announcements for this seminar, as well as for related seminars and events, are sent to the GT seminar mailing list. You can subscribe directly or by contacting Walter Neumann.